Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- Part I Overview and background topics
- Part II Density functional theory
- Part III Important preliminaries on atoms
- Part IV Determination of electronic structure: the three basic methods
- Part V Predicting properties of matter from electronic structure – recent developments
- Appendix A Functional equations
- Appendix B LSDA and GGA functionals
- Appendix C Adiabatic approximation
- Appendix D Response functions and Green's functions
- Appendix E Dielectric functions and optical properties
- Appendix F Coulomb interactions in extended systems
- Appendix G Stress from electronic structure
- Appendix H Energy and stress densities
- Appendix I Alternative force expressions
- Appendix J Scattering and phase shifts
- Appendix K Useful relations and formulas
- Appendix L Numerical methods
- Appendix M Iterative methods in electronic structure
- Appendix N Code for empirical pseudopotential and tight-binding
- Appendix O Units and conversion factors
- References
- Index
Appendix B - LSDA and GGA functionals
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- Part I Overview and background topics
- Part II Density functional theory
- Part III Important preliminaries on atoms
- Part IV Determination of electronic structure: the three basic methods
- Part V Predicting properties of matter from electronic structure – recent developments
- Appendix A Functional equations
- Appendix B LSDA and GGA functionals
- Appendix C Adiabatic approximation
- Appendix D Response functions and Green's functions
- Appendix E Dielectric functions and optical properties
- Appendix F Coulomb interactions in extended systems
- Appendix G Stress from electronic structure
- Appendix H Energy and stress densities
- Appendix I Alternative force expressions
- Appendix J Scattering and phase shifts
- Appendix K Useful relations and formulas
- Appendix L Numerical methods
- Appendix M Iterative methods in electronic structure
- Appendix N Code for empirical pseudopotential and tight-binding
- Appendix O Units and conversion factors
- References
- Index
Summary
Summary
In this appendix are given representative forms for the exchange–correlation energy and potential in the LSDA and GGA approximations. The forms given here are chosen because they are widely used and are relatively simple. Actual programs that provide energies and potentials for these and other forms can be found on-line (see Ch. 24).
Local spin density approximation (LSDA)
The local density approximation is based upon the exact expressions for the exchange energy, Eq. (5.15), and various approximations and fitting to numerical correlation energies for the homogeneous gas. Comparison of the forms is shown in Fig. 5.4. The first functions were the Wigner interpolation formula, Eq. (5.22), and the Hedin–Lundqvist [220] form; the latter is derived from many-body perturbation theory and is given below. As described in Ch. 5, the quantum Monte Carlo (QMC) calculations of Ceperley and Alder [297], and more recent work [298, 299, 303] provide essentially exact results for unpolarized and fully polarized cases. These results have been fitted to analytic forms for εc(rs), where rs is given by Eq. (5.1), leading to two widely used functionals due to Perdew and Zunger (PZ) [300] and Vosko, Wilkes, and Nusiar (VWN) [301], which are very similar quantitatively. Both functionals assume an interpolation form for fractional spin polarization, and Ortiz and Balone [298] report that their QMC calculations at intermediate polarization are somewhat better described by the VWN form.
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- Electronic StructureBasic Theory and Practical Methods, pp. 479 - 481Publisher: Cambridge University PressPrint publication year: 2004